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Instability of Resonances Under Stark Perturbations

  • Arne JensenEmail author
  • Kenji Yajima
Article
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Abstract

Let \(H^{\varepsilon }=-\frac{\mathrm{d}^2}{\mathrm{d}x^2}+\varepsilon x +V\), \(\varepsilon \ge 0\), on \(L^2(\mathbf {R})\). Let \(V=\sum _{k=1}^Nc_k|{\psi _k}\rangle \langle {\psi _k}|\) be a rank N operator, where the \(\psi _k\in L^2(\mathbf {R})\) are real, compactly supported, and even. Resonances are defined using analytic scattering theory. The main result is that if \(\zeta _n\), \({{\,\mathrm{Im}\,}}\zeta _n<0\), are resonances of \(H^{\varepsilon _n}\) for a sequence \(\varepsilon _n\downarrow 0\) as \(n\rightarrow \infty \) and \(\zeta _n\rightarrow \zeta _0\) as \(n\rightarrow \infty \), \({{\,\mathrm{Im}\,}}\zeta _0<0\), then \(\zeta _0\) is not a resonance of \(H^0\).

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Notes

Acknowledgements

KY thanks Ira Herbst for asking him about the instability of resonances under Stark perturbations. KY is supported by JSPS grant in aid for scientific research No. 16K05242. AJ acknowledges support from the Danish Council of Independent Research | Natural Sciences, Grant DFF4181-00042.

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© Springer Nature Switzerland AG 2018

Authors and Affiliations

  1. 1.Department of Mathematical SciencesAalborg UniversityAalborg ØDenmark
  2. 2.Department of MathematicsGakushuin UniversityTokyoJapan

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